One way of determining the concentration of a substance in a sample is by performing serial dilution on the sample. Serial dilution techniques collect a finite number of data points for the sample by taking one or more observations (e.g., indicating optical density) of various dilutions (e.g., dilutions formed by adding various quantity of diluent to the sample). For example, dilutions of 10%, 1%, 0.1%, etc. can be measured for optical density.
The results can then be used to determine a concentration of the substance in the sample via reference to a sigmoid curve representing serial-dilution observations for a sample having a known concentration of the substance (sometimes called a “standard” or “characteristic” sigmoid curve). FIG. 1 shows such a sigmoid curve 120. The sigmoid curve can be represented by the four-parameter Formula (1).
                              f          ⁡                      (            x            )                          =                              β            2                    +                                                    β                1                            -                              β                2                                                    1              +                                                (                                      x                                          β                      3                                                        )                                                  β                  4                                                                                        (        1        )            The parameters of Formula (1) can be chosen so that the function ƒ(x) calculates the optical density based on a particular dilution x. Given an optical density 130 for a sample having an unknown concentration of the substance and the degree of dilution associated with the sample, the concentration of the substance can be back-calculated. In practice, plural observations of the optical density can be taken for plural degrees of dilution and applied to the standard curve.
Various techniques have been used to define the curve, analyze the observations, and calculate a concentration. One method is described by O'Connell, et al., “Calibration and assay development using the four-parameter logistic model,” Chemometrics and Intelligent Laboratory Systems, 20 (1993) 97-114, Elsevier Science Publishers B.V., Amsterdam (“O'Connell”), which is hereby incorporated herein by reference. The O'Connell approach describes determining a minimum detectable concentration (MDC) and a reliable detection limit (RDL).
The O'Connell technique can produce significant variability in its results. In certain scenarios, variability is to be avoided. Therefore, there exists a need for technologies that avoid variability in their results and otherwise make better use of the sigmoid curve.